When playing Evelyn in the laning phase you have to be incredibly smart. You do great damage early but it wanes off then you should become tanky late game. You do well with early game powerhouse ADC's, such as Corki, Graves, or Lucian. High mobility ADC's are good as well. You can't be hesitant as well, if you are you'll miss your chance which might lead to a dead ADC or not dead enemy ADC. Roam if you are far ahead, the enemy bot lane will play safe, expecting that you are somewhere around.

Beware, your mid game sucks, you don't do great damage and you aren't incredibly tanky. In team fights though you should aim to flank and target their squishy characters. Just the threat of you, your ult, and ignite will get them to turn their focus toward you which should give your carries a chance to do some damage. Late game team fights you should be tanky enough to not instantly die once everyone focuses you. A well timed ult is important.

-An anti engage type support
-Passive causes entire enemy team to play differently
-Strong early game
-Ok late game
-A very utility based late game
-great synergy with early game ADC's

Con's
-Bad mid game
-Only OK late game
-Not a true tank
-Isn't as effective if losing lane